Maximum Likelihood Estimation for Stochastic Differential Equations Using Sequential Gaussian-Process-Based Optimization

References

• Aït-Sahalia, Y. (2002), “Maximum Likelihood Estimation of Discretely Sampled Diffusions: A Closed-form Approximation Approach,” Econometrica, 70, 223–262.
   Web of Science ®Google Scholar
• ——— (2008), “Closed-Form Likelihood Expansions for Multivariate Diffusions,” The Annals of Statistics, 36, 906–937.
   Web of Science ®Google Scholar
• Bally, V., and Talay, D. (1996), “The Law of the Euler Scheme for Stochastic Differential Equations (II): Convergence Rate of the Density,” Monte Carlo Methods and Applications, 2, 93–128.
   Google Scholar
• Beskos, A., Papaspiliopoulos, O., and Roberts, G. O. (2006a), “Retrospective Exact Simulation of Diffusion Sample Paths With Applications,” Bernoulli, 12, 1077–1098.
   Web of Science ®Google Scholar
• ——— (2008), “A Factorisation of Diffusion Measure and Finite Sample Path Constructions,” Methodology and Computing in Applied Probability, 10, 85–104.
   Web of Science ®Google Scholar
• Beskos, A., Papaspiliopoulos, O., Roberts, G. O., and Fearnhead, P. (2006b), “Exact and Computationally Efficient Likelihood-Based Estimation for Discretely Observed Diffusion Processes,” Journal of the Royal Statistical Society, Series B, 68, 333–382.
   Google Scholar
• Beskos, A., and Roberts, G. O. (2005), “Exact Simulation of Diffusions,” The Annals of Applied Probability, 15, 2422–2444.
   Web of Science ®Google Scholar
• Bladt, M., and Sørensen, M. (2014), “Simple Simulation of Diffusion Bridges With Application to Likelihood Inference for Diffusions,” Bernoulli, 20, 645–675.
   Web of Science ®Google Scholar
• Brandt, M. W., and Santa-Clara, P. (2002a), “Numerical Techniques for Maximum Likelihood Estimation of Continuous-time Diffusion Processes: Comment,” Journal of Business & Economic Statistics, 20, 321–324.
   Google Scholar
• ——— (2002b), “Simulated Likelihood Estimation of Diffusions With an Application to Exchange Rate Dynamics in Incomplete Markets,” Journal of Financial Economics, 63, 161–210.
   Web of Science ®Google Scholar
• Chan, K. C., Karolyi, G. A., Longstaff, F. A., and Sanders, A. B. (1992), “An Empirical Comparison of Alternative Models of the Short-Term Interest Rate,” The Journal of Finance, 47, 1209–1227.
   Web of Science ®Google Scholar
• Craigmile, P. F., Cressie, N., Santner, T., and Rao, Y. (2006), “A Loss Function Approach to Identifying Environmental Exceedances,” Extremes, 8, 143–159.
   Google Scholar
• Cressie, N., and Pavlicová, M. (2005), “Lognormal Kriging: Bias Adjustment and Kriging Variances,” in Geostatistics Banff 2004, eds. O. Leuangthong and V. Deutsch, New York: Springer, pp. 1027–1036.
   Google Scholar
• Cressie, N., Stern, H., and Wright, D. (2000), “Mapping Rates Associated With Polygons,” Journal of Geographical Systems, 2, 61–69.
   Google Scholar
• Cressie, N., and Wikle, C. K. (2011), Statistics for Spatio-Temporal Data, New York: Wiley.
   Google Scholar
• Del Moral, P., and Murray, L. M. (2015), “Sequential Monte Carlo With Highly Informative Observations,” SIAM/ASA Journal of Uncertainty Quantification, 3, 969–997.
   Google Scholar
• Durham, G. B., and Gallant, A. R. (2002), “Numerical Techniques for Maximum Likelihood Estimation of Continuous-time Diffusion Processes,” Journal of Business & Economic Statistics, 20, 297–316.
   Web of Science ®Google Scholar
• Eddelbuettel, D. (2011), “Seamless R and C++ Integration With Rcpp,” Journal of Statistical Software, 40. doi: 10.18637/jss.v040.i08.
   Web of Science ®Google Scholar
• Eddelbuettel, D., and Sanderson, C. (2014), “RcppArmadillo: Accelerating R With High-performance C++ Linear Algebra,” Computational Statistics and Data Analysis, 71, 1054–1063.
   Web of Science ®Google Scholar
• Elerian, O., Chib, S., and Shephard, N. (2001), “Likelihood Inference for Discretely Observed Nonlinear Diffusions,” Econometrica, 69, 959–993.
   Web of Science ®Google Scholar
• Gramacy, R. B., and Lee, H. K. (2008), “Bayesian Treed Gaussian Process Models With an Application to Computer Modeling,” Journal of the American Statistical Association, 103, 1119–1130.
   Web of Science ®Google Scholar
• Herbei, R., and Berliner, L. M. (2014), “Estimating Ocean Circulation: An MCMC Approach With Approximated Likelihoods via the Bernoulli Factory,” Journal of the American Statistical Association, 109, 944–954.
   Web of Science ®Google Scholar
• Herbei, R., McKeague, I. W., and Speer, K. G. (2008), “Gyres and Jets: Inversion of Tracer Data for Ocean Circulation Structure,” Journal of Physical Oceanography, 38, 1180–1202.
   Web of Science ®Google Scholar
• Higdon, D., Gattiker, J., Williams, B., and Rightley, M. (2008), “Computer Model Calibration Using High-Dimensional Output,” Journal of the American Statistical Association, 103, 570–583.
   Web of Science ®Google Scholar
• Hogg, N. G., and Owens, W. B. (1999), “Direct Measurement of the Deep Circulation Within the Brazil Basin,” Deep Sea Research Part II: Topical Studies in Oceanography, 46, 335–353.
   Web of Science ®Google Scholar
• Hurn, A. S., Jeisman, J. I., and Lindsay, K. A. (2007), “Seeing the Wood for the Trees: A Critical Evaluation of Methods to Estimate the Parameters of Stochastic Differential Equations,” Journal of Financial Econometrics, 5, 390–455.
   Web of Science ®Google Scholar
• Iacus, S. M. (2009), Simulation and Inference for Stochastic Differential Equations: With R Examples (Vol. 1), New York: Springer.
   Google Scholar
• Jones, D. R., Schonlau, M., and Welch, W. J. (1998), “Efficient Global Optimization of Expensive Black-Box Functions,” Journal of Global Optimization, 13, 455–492.
   Web of Science ®Google Scholar
• Karatzas, I., and Shreve, E. S. (1991), Brownian Motion and Stochastic Calculus, New York: Springer Science & Business Media.
   Google Scholar
• Kennedy, M. C., and O’Hagan, A. (2001), “Bayesian Calibration of Computer Models,” Journal of the Royal Statistical Society, Series B, 63, 425–464.
   Web of Science ®Google Scholar
• Kloeden, P. E., and Platen, E. (1992), Numerical Solution of Stochastic Differential Equations, New York: Springer.
   Google Scholar
• Lin, M., Chen, R., and Mykland, P. (2010), “On Generating Monte Carlo Samples of Continuous Diffusion Bridges,” Journal of the American Statistical Association, 105, 820–838.
   Web of Science ®Google Scholar
• Lo, A. W. (1988), “Maximum Likelihood Estimation of Generalized Itô Processes With Discretely Sampled Data,” Econometric Theory, 4, 231–247.
   Web of Science ®Google Scholar
• McKeague, I. W., Nicholls, G., Speer, K., and Herbei, R. (2005), “Statistical Inversion of South Atlantic Circulation in an Abyssal Neutral Density Layer,” Journal of Marine Research, 63, 683–704.
   Web of Science ®Google Scholar
• Papaspiliopoulos, O., and Roberts, G. (2012), “Importance Sampling Techniques for Estimation of Diffusion Models,” Statistical Methods for Stochastic Differential Equations, 124, 311–340.
   Google Scholar
• Pedersen, A. R. (1995), “A New Approach to Maximum Likelihood Estimation for Stochastic Differential Equations Based on Discrete Observations,” Scandinavian Journal of Statistics, 22, 55–71.
   Web of Science ®Google Scholar
• Rasmussen, C. E., and Williams, C. K. I. (2006), Gaussian Processes for Machine Learning, Cambridge, MA: The MIT Press.
   Google Scholar
• R Core Team (2013), R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing.
   Google Scholar
• Roberts, G. O., and Stramer, O. (2001), “On Inference for Partially Observed Nonlinear Diffusion Models Using the Metropolis-Hastings Algorithm,” Biometrika, 88, 603–621.
   Web of Science ®Google Scholar
• Sacks, J., Welch, W. J., Mitchell, T. J., and Wynn, H. P. (1989), “Design and Analysis of Computer Experiments,” Statistical Science, 4, 409–423.
   Google Scholar
• Santa-Clara, P. (1997), “Simulated Likelihood Estimation of Diffusions With an Application to the Short Term Interest Rate,” Technical Report, Anderson Graduate School of Management, UCLA.
   Google Scholar
• Santner, T. J., Williams, B. J., and Notz, W. I. (2003), The Design and Analysis of Computer Experiments, New York: Springer.
   Google Scholar
• Schonlau, M., Welch, W. J., and Jones, D. R. (1998), “Global versus Local Search in Constrained Optimization of Computer Models,” in New Developments and Applications in Experimental Design, eds. N. Flournoy, W. F. Rosenberger, and W. K. Wong, Hayward, CA: Institute of Mathematical Statistics, pp. 11–25.
   Google Scholar
• Stein, M. L. (1999), Interpolation of Spatial Data: Some Theory for Kriging, New York: Springer-Verlag.
   Google Scholar
• Stramer, O., and Yan, J. (2007a), “Asymptotics of an Efficient Monte Carlo Estimation for the Transition Density of Diffusion Processes,” Methodology and Computing in Applied Probability, 9, 483–496.
   Web of Science ®Google Scholar
• ——— (2007b), “On Simulated Likelihood of Discretely Observed Diffusion Processes and Comparison to Closed-Form Approximation,” Journal of Computational and Graphical Statistics, 16, 672–691.
   Web of Science ®Google Scholar
• Uhlenbeck, G. E., and Ornstein, L. S. (1930), “On the Theory of the Brownian Motion,” Physical Review, 36, 823–841.
   Google Scholar
• Van Der Vaart, A., and Van Zanten, H. (2011), “Information Rates of Nonparametric Gaussian Process Methods,” The Journal of Machine Learning Research, 12, 2095–2119.
   Web of Science ®Google Scholar
• Williams, B. J., Santner, T. J., and Notz, W. I. (2000), “Sequential Design of Computer Experiments to Minimize Integrated Response Functions,” Statistica Sinica, 10, 1133–1152.
   Web of Science ®Google Scholar
• Wright, D., Stern, H., and Cressie, N. (2003), “Loss Functions for Estimation of Extrema With an Application to Disease Mapping,” The Canadian Journal of Statistics, 31, 251–266.
   Web of Science ®Google Scholar
• Wunsch, C. (1996), The Ocean Circulation Inverse Problem, Cambridge, England: Cambridge University Press.
   Google Scholar
• Zhang, H. (2004), “Inconsistent Estimation and Asymptotically Equal Interpolations in Model-Based Geostatistics,” Journal of the American Statistical Association, 99, 250–261.
   Web of Science ®Google Scholar
• Zhang, J., Craigmile, P. F., and Cressie, N. (2008), “Loss Function Approaches to Predict a Spatial Quantile and its Exceedance Region,” Technometrics, 50, 216–227.
   Web of Science ®Google Scholar
• Zimmerman, D. L., and Zimmerman, M. B. (1991), “A Comparison of Spatial Semivariogram Estimators and Corresponding Ordinary Kriging Predictors,” Technometrics, 33, 77–91.
   Web of Science ®Google Scholar